In [1]}, Gay and Kirby proved that every closed smooth 4-manifold admits a trisection. A trisection of a closed 4-manifold \(X\)  is a decomposition \(X=X_1\cup X_2\cup X_3\)  into three 4-dimensional 1-handlebodies so that the pairwise intersections are 3-dimensional 1-handlebodies \(X_i\cap X_j\) , and the triple intersection is a closed surface \(\Sigma =X_1\cap X_2\cap X_3\) .
The genus of the trisection is defined as the genus of \(\Sigma \) .
In recent years, the notion of trisections has been extended to 4-manifolds with several boundaries [2]}, knotted surfaces in 4-manifolds [3]} and finitely presented groups [4]}.
See [5]} for an exposition on recent advances in the theory of trisections of 4-manifolds.
